A control system typically drives a controlled system, i.e., a plant, such that the plant responds in a desired manner; the response of the plant is often called the plant's output response. For example, an elevator-control system typically drives an elevator motor such that the attached elevator accelerates and decelerates smoothly and stops without oscillation and with its floor even with the lobby floor. Also, a cruise-control system controls the engine of an automobile such that the automobile accelerates or decelerates smoothly and without oscillation to a preset speed and then maintains the preset speed.
Negative feedback is a popular control-system topology that forces a plant to have a desired output response. A first order negative-feedback control system, also known as a closed-loop control system, generates a negative-feedback signal having a magnitude that is related to the magnitude of the plant's actual output response and having a phase that is opposite to—ideally 180° out of phase with—the phase of the input signal. The control system typically sums the negative-feedback signal with the input signal to generate a signal for driving the plant. The difference between the plant's actual and desired output responses is the system error, and the negative-feedback signal operates to reduce the system error toward zero and to below a predetermined threshold. How close to zero the negative-feedback signal can reduce the system error is inversely proportional to the closed-loop gain of the control system. Furthermore, the speed at which the negative feedback signal can reduce the system error to below the predetermined threshold is proportional to the closed-loop bandwidth of the control system. For example, suppose that when an automobile is traveling at 60 miles per hour (MPH) a driver engages the cruise-control system and sets the cruise speed to 70 MPH. Because the actual speed of 60 MPH is 10 MPH less than the desired speed of 70 MPH, the error is −10 MPH, and the negative-feedback signal causes the automobile to accelerate to 70 MPH. As the speed of the automobile increases above 60 MPH, the error decreases. When the error decreases to within a predetermined error range, such as ±1 MPH, the negative-feedback signal causes the automobile to maintain its speed within corresponding predetermined range, such as 69-71 MPH. The predetermined error range is inversely proportional to the control system's close-loop gain, and the time that the negative-feedback control system requires to accelerate the speed of the automobile from 60 MPH to 70 MPH is proportional to the control-system's closed-loop bandwidth. That is, a designer can reduce the predetermined error range by increasing the closed-loop gain of the negative-feedback control system and can increase the rate of acceleration by increasing the closed-loop bandwidth of the control system. But if a designer increases the closed-loop bandwidth or gain too much, then the negative-feedback control system will be unstable and will cause the speed of the automobile to oscillate.
FIG. 1 is a plan view of a conventional microelectromechanical scanning-mirror (MEMS) assembly 10, which includes a reflector 12, vertical-sweep torsion arms 16a and 16b, horizontal-sweep torsion arms 18a and 18b, and a base 20 to which the vertical-sweep torsion arms are anchored. The reflector 12 and torsion arms 16a and 16b collectively form a vertical-sweep plant, i.e., a vertical beam scanner. The assembly 10 also includes vertical and horizontal sweep mechanisms (not shown) for rotating the reflector 12 about the vertical-sweep torsion arms 16a and 16b and the horizontal-sweep torsion arms 18a and 18b, respectively. By sweeping a light beam 22 in the horizontal and vertical dimensions, the reflector 12 scans a viewable image (not shown) on a display screen (not shown) or onto a viewer's retina (not shown). A MEMS assembly like the assembly 10 is disclosed in U.S. Pat. No. 5,629,790, entitled MICROMACHINED TORSIONAL SCANNER, issued May 13, 1997, to Armand P. Neukermans et al., which is incorporated by reference.
In operation, a vertical drive signal (not shown) causes the vertical sweep mechanism (not shown) to rotate the reflector 12 back and forth about the arms 16a and 16b, and to thus sweep the beam 22 in the vertical dimension. Simultaneously, a horizontal drive signal (not shown) causes the horizontal-sweep mechanism (not shown) to rotate the reflector 12 back and forth about the arms 18a and 18b, and to thus sweep the beam 22 in the horizontal dimension.
FIG. 2 is a time-domain plot of a desired, i.e., target, output response of the reflector 12 of FIG. 1 in the vertical dimension. The vertical axis represents the vertical position of the sweep beam 22 (FIG. 1), and the horizontal axis represents time. The output response, often called a “sawtooth wave,” is periodic, with a period Tscan, which may equal, for example, 1/60th of a second. At time t0, the beam 22 is at the bottom of the scanned image (not shown), and thus at its minimum amplitude −A. −A corresponds to a maximum rotation of the reflector 12 about the torsion arms 16a and 16b in a first direction for the depicted application. During a flyback period Tflyback between times t0 and t1, the beam 22 is inactive and rapidly “flies back” to the top of the scanned image, and thus to its maximum amplitude +A. +A corresponds to a maximum rotation of the reflector 12 about the torsion arms 16a and 16b in a second direction for the depicted application, the second direction being opposite to the first direction. During a sweep period Tsweep between the times t1 and t2, the beam 22 is active and the reflector 12 sweeps the beam from its maximum vertical amplitude +A to its minimum vertical amplitude −A at a constant rate, i.e., constant velocity, to provide a constant vertical spacing between adjacent pairs of the horizontal lines that compose the scanned image (not shown).
FIG. 3 is a time-domain plot of a possible actual output response of the reflector 12 of FIG. 1 in the vertical dimension. This actual output response includes unwanted transient oscillations, often called “ringing”, between times t1 and t3. Consequently, the reflector 12 is not sweeping the beam 22 (FIG. 1) at a constant vertical velocity, and is thus causing uneven vertical spacing between adjacent horizontal lines of the scanned image (not shown), during this transient period. Unfortunately, such uneven line spacing may generate visible artifacts such as “banding” (alternative light and dark horizontal bands) in the scanned image.
One technique for forcing the actual vertical sweep output response of the reflector 12 close enough to the target output response of FIG. 2 to eliminate image artifacts caused by uneven line spacing is driving the reflector with a negative-feedback control system as described above.
But unfortunately, even when driven with a negative-feedback control system, the reflector 12 may not have an actual output response that is close enough to the target output response of FIG. 2 to eliminate such image artifacts. A negative-feedback control system typically requires a relatively high gain and a relatively wide bandwidth to cause the reflector 12 to have an output response close to that of FIG. 2. But a high gain or wide bandwidth may give the control system a low margin of stability as discussed above. Furthermore, a wide bandwidth may cause the control system to amplify noise, i.e., to have a low noise immunity. A low stability margin may hinder the mass production of an image system that incorporates the reflector 12, because variations in the reflectors and other system components, even within expected tolerance ranges, may render some of the systems unstable, and thus unusable. And low noise immunity may allow noise-induced jitter or other disturbances to adversely affect the rotation of the reflector 12 in the vertical dimension, and thus may cause the actual output response of the reflector to deviate from the target output response by an unacceptable amount.